Undrained shear behavior of silty sand with a constant state parameter considering initial stress anisotropy effect

Field observations in sedimentation and erosion-prone areas indicate that most natural sand deposits may contain a certain amount of non-plastic fines and are often under anisotropic stress conditions. A series of triaxial compression tests were performed on clean and silty sand with fines content fc ranging from 0 to 20% at an initial mean effective stress of p0′ = 100 kPa and varying consolidation conditions to understand the impact of initial stress anisotropy on undrained shear behavior. The results indicate that the state parameter ψ is a superior predictor for characterizing the responses of sand-fines mixtures compared to the global void ratio and relative density. A comparison of the behavior of clean and silty sand with a constant ψ (= − 0.03) confirms that the sample with 10% fc exhibits the strongest dilation and greatest shear resistance, irrespective of the consolidation conditions. It is also demonstrated that the initial stress anisotropy with a comparably higher static stress ratio ηs typically diminishes the shear strength of mixtures. However, the influence of initial stress anisotropy on soil stiffness is not unilateral. The sample consolidated to a negative ηs is stiffer than that under isotropic consolidation, while the presence of a positive ηs leads to a decrease in the secant Young's modulus.


Materials and methods
The material tested in this study was Fujian sand, a type of Chinese standard sand composed of sub-angular to sub-rounded silica grains.The sand grains were ground into non-plastic fines with particle diameters primarily ranging from 2 to 75 μm.The particle size distribution curves and basic physical properties of the test materials are shown in Fig. 1 and Table 1, respectively.The silty sand sample that is formed by mixing crushed fines with the host sand was controlled at f c = 5%, 10%, and 20%.The maximum void ratio (e max ) was determined using Method B in ASTM D4254-16 48 with a cylindrical tube, and the minimum void ratio (e min ) was measured through vibratory table tests 49 .Although the procedures were recommended for samples with small fines fractions, they were extended in testing mixed soils at a higher f c to achieve a consistent comparison 11,15 .The dry proctor test method 7,11 was also adopted to measure e min of the silty sand with f c > 10% in the present study.In www.nature.com/scientificreports/agreement with the findings of Polito and Martin 15 , the proctor test yields a similar e min value to that produced by the vibration method.The results show that the limiting void ratios first decrease and then increase with increasing f c , and the lowest values appear at around f c = 30%, which can be roughly regarded as f c,th .This value is comparable with that of f c,th = 29% determined from the empirical approach using Eq.(1), which has been verified and widely adopted 10,50 .
Tests were performed utilizing an automated triaxial testing system, as described by Zhou et al. 51 .Samples of approximately 50 mm in diameter and 100 mm in height were prepared via the moist tamping method, which avoids segregation between sand and fine particles 26,52 .The oven-dried sand, mixed with de-aired water to achieve 5% moisture content, was compacted in three layers into a split mold using a small hammer.After removing the split mold, the actual dimensions of the sample and its volumetric strain during preparation and consolidation were measured to calculate the actual density state, which is also determined through the back-analysis procedure by measuring the final water content of the sample at the end of tests 53 .The samples were completely saturated with Skempton's B-values above 0.95 and then isotropically or anisotropically consolidated to a specific static deviatoric stress q s (= σ v − σ h , where σ v and σ h are the vertical and horizontal normal stress, respectively) at an initial mean effective stress of p 0 ' = 100 kPa.The controlled void ratio after consolidation is achieved by a trial procedure.To achieve the same range of void ratio as samples undergoing isotropic consolidation, the sample that will be anisotropically consolidated is intended to be densely or loosely deposited at the preparation stage, depending on its volume change tendency during anisotropic consolidation.If the void ratio of a sample is still out of the desired range, it will be discarded.In fact, the test series selected and shown below are the best outcome in terms of the samples' density for comparison purpose.Then strain-controlled triaxial compression tests were conducted under undrained conditions by applying the monotonic deviatoric stress at a strain rate of 0.1%/ min, which was widely adopted by Hyodo et al. 31 , Murthy et al. 6 , Pan et al. 54 , and Porcino et al. 55 to evaluate the undrained shear response of both clean and silty sand.
Table 2 summarizes all the monotonic triaxial conditions investigated in this study, as designated by state indices (e, D r , and ψ), stress variables (σ v , σ h , and q s ), and stress anisotropy factor η s (= q s /p 0 '), which are classified into four series based on the controlled f c .The results obtained from the benchmark tests in each series were compared to preliminarily examine the suitability of e, D r , and ψ in characterizing the undrained shear behavior of isotropically consolidated samples.Subsequently, tests addressing different η s levels were performed on anisotropically consolidated samples under a constant ψ (= − 0.03) condition.www.nature.com/scientificreports/dilation, whereas the sample with e = 0.845 exhibits an initial contractive phase, followed by dilative behavior.

Undrained shear behavior of isotropically consolidated samples
In between is the case of e = 0.826, in which the phase-transformation state 56 that divides the contractive and dilative responses can also be found.In Fig. 3a, all the three clean sand samples display stable strain-hardening behavior toward the critical state.Similarly, for the silty sand a given f c , a decrease in e leads to a shift from a tendency to contract to a tendency to dilate (Fig. 2), accompanied by a transition from strain-softening to strainhardening behavior (Fig. 3). Figure 4 presents a comparison between the responses of clean and silty sand under isotropic consolidation.The basis for the comparison of samples in Fig. 4a,b is the post-consolidation relative density (D r = 18.9%).It is shown that the addition of silty fines results in a decrease in shear strength and an increase in the degree of strain softening compared to the clean sand, although the void ratio declines with increasing f c .When comparing the responses of samples having a constant e (= 0.782), as shown in Fig. 4c,d, the clean sand also exhibits a very dilative response, with a marked and stable increase in strength compared with that of silty sand.In fact, the behavior of silty sand at a lower f c is primarily controlled by the sand matrix because the silty fines may fill the voids formed by coarse grains with less participation in the force transfer mechanism.
In most of the tests conducted in this study, the axial strain developed over 25% and the rate of variation in deviatoric stress at that strain level was relatively small; such a state is postulated close enough and can be used to approximate the critical state.The critical state data for clean and silty sand are displayed on q-p' and e-logp' planes in Fig. 5a,b, respectively.The plot shows that the critical state stress points in q-p' plane fall within a narrow band that can be represented by straight lines passing through the origin where M cs is the critical state stress ratio relating to the critical state friction angle ϕ cs as A detailed scrutiny of Fig. 5a shows that the clean Fujian sand has a ϕ cs value of 30.9°, being comparable with that obtain from Yang and Wei 52 on the same sand and less than the value for Toyoura sand that is more angular.For the sand mixed with 5-20% fines, Fig. 5a shows a slightly higher ϕ cs of 32.1°, although it appears insensitive to the increasing fines contents.This is mainly due to the presence of crushed fines that are angular with irregular geometry and implies that the friction angle of a mixed soil is affected by the shape of both coarse and finer particles.Thus, the shape characteristics of the tested host sand and fines, including the aspect ratio, sphericity, and roundness, should be examined in a quantitative way and further compared with other types of finer and coarse grains in future, which may allow a better understanding of their macro-scale properties that are potentially associated with the particle shape.
Compared with the CSL in the q-p' plane that is insensitive to f c variation, the critical state locus in the e-logp' plane shown in Fig. 5b descends as f c increases up to 20%.This observation aligns with that reported by Thevanayagam et al. 22 , Murthy et al. 6 , and Yang et al. 25 on several different silty sands.Of note, the critical state locus on the semi-log form is not a straight line but instead a curved line that can be described through a power function of the form proposed by Li and Wang 57 where e cs is the critical state void ratio, p a denotes atmospheric pressure (101 kPa), e Γ , λ, and ξ are fitting parameters.The least-square regression is employed to estimate the expression for each mixture's CSL (Fig. 5b), assuming that the power-law exponent ξ is constant and equal to 0.6 in line with Yang et al. 's 25 finding.The parameters e Γ and λ exhibit a decreasing trend with increasing f c .Some scatter exists in the critical state data, especially when f c goes up, which may be contributed to the change of achieving uniform sample with increasing f c and the inherent variability in the material.Ni et al. 2 and Chiu and Fu 29 have noted that further increase in f c might lead to an upward movement of the CSL.This signifies the presence of the threshold f c,th distinguishing the locations of the CSL of silty sand, which is beyond the range of f c (0-20%) considered this study.
The behavior of clean and silty sand samples that are packed at a constant ψ is shown in Fig. 6, where the value of ψ = − 0.03 is calculated using the CSL specific to f c of the sand-fines mixtures, which provides convenience in controlling of samples' density under the testing conditions.Selecting this value allows the attainment of samples in a loose or medium dense state (D r ≈ 20-50%, as shown in Table 2), neither too loose nor too dense, facilitating subsequent experimental operations and data acquisition.All four isotropically consolidated www.nature.com/scientificreports/samples that are undrained sheared from a negative ψ exhibit a prominent dilative, stable strain-hardening behavior, which conforms with the framework of critical state soil mechanics.Specifically, the clean sand behaves a contractive phase at the start of undrained shearing, while the degree of contraction decreases as f c increases and the sample with f c = 10% shows a fully dilative behavior.A transition arises when f c increases to 20%, leading to a slightly contractive initial phase for the sample.Correspondingly, the shear strength at the critical state increases first and then decreases with increasing f c , with the sample at 10% f c manifesting the highest shear strength.As granular materials in nature, the sand-silt mixtures comprising discrete particles exhibit complex structure and mechanical behaviors during the loading process.Shire et al. 58,59 investigated the micro-structure and micro-properties of granular mixtures under isotropic compression.It has been shown that for a mixture containing a comparatively lower f c , the finer grains fill voids left by the coarse ones without separating them, such that the latter sustaining the strong force chains will be significantly reinforced by a large number of fines around them.As f c further increases to a critical fraction, some of the fines tend to separate the coarse particles from one another, leading to the soil matrix being less resistant to the external forces.This may account for the above phenomenon that samples with 10% and 20% f c behave a notable deviation in the stress-strain behavior.
Compared with the dramatic differences observed in soil behavior under constant e or D r (Fig. 4), the overall analogous behavior shown in Fig. 6 implies that ψ is a comparatively superior predictor for synthesizing the undrained shear behavior of clean and silty sand.The instability state (IS) representing the peak stress point is a striking feature associated with the strainsoftening behavior of sand.In Fig. 7, the stress ratio at the onset of IS, η IS = q IS /p IS ', is plotted against ψ; also included are the data from Yang and Wei 52 on the similar test material for comparison.The variations in data points indicate that the trend of η IS with respect to ψ is sensitive to the amount of fines.Nevertheless, there exist fairly good correlations between η IS and ψ, in accordance with which η IS decreases with ψ reported by Yang and Wei 52 , Lashkari et al. 35 , and Fakharian et al. 38 , implying that the instability is triggered at a lower stress ratio for samples with higher contractive tendency.

Effect of initial stress anisotropy on strength and stiffness characteristics
The above results demonstrate the effectiveness of the constant ψ approach in the comparative study of the soil mixtures having f c values ranging from 0 to 20%.Therefore, the following interpretations regarding the initial stress anisotropy effect are based on tests conducted under the same ψ value of − 0.03.Figures 8, 9 and 10 present the effective stress paths and stress-strain curves of clean and silty sand samples that are anisotropically  www.nature.com/scientificreports/consolidated to different η s .In Fig. 8, the clean sand subjected to η s = 0.4 behaves a contraction-to-dilation manner, while samples mixed with 5% and 10% f c show a more dilative behavior.At f c = 20%, the sample displays a highly contractive response and achieves a transient minimum shear strength known as the quasi-steady state 60 prior to dilation and strain hardening toward the critical state.The scenario that the sample with 10% f c has the strongest dilation and highest strength is compatible with that observed in isotropically consolidated samples   www.nature.com/scientificreports/(Fig. 6).Similar observations can be drawn for the sample subjected to η s = 0.6 and − 0.4, as depicted in Figs. 9 and 10, respectively.Note that the situation where η s = − 0.4 corresponds to an initial extensional static stress, which is a less investigated but particular point of interest 36 .Moreover, it is found that a comparably higher η s level leads to a deterioration in the monotonic shear behavior of the sand-silt mixtures.For example, Fig. 9 shows that the silty sand with f c = 20% that undergoes anisotropic consolidation to η s = 0.6 is characterized by fully contractive and strain-softening behaviors.This type of behavior, typically associated with the instability, is commonly referred to as static liquefaction 61 .Figure 11 further presents the stress strain curve of samples with varying degrees of initial stress anisotropy but with a constant state parameter (ψ = − 0.03).Regardless of f c levels (0-20%) considered herein, the sample under isotropic consolidation (η s = 0) exhibits a marked stable strain-hardening behavior, in comparison to that undergoes initial stress anisotropy.More specifically, Fig. 11a,d clearly show that samples behave from a strainhardening type response to a strain-softening type response, as the degree of stress anisotropy increases.The presence of the initial stress anisotropy also has a weakened effect on the undrained shear strength, as illustrated in detail in the following discussion.The undrained shear strength represented by q u is determined by the peak deviatoric stress for strain-softening type response or the mobilized deviatoric stress at a strain level of 15% for strain-hardening type response.Figure 12 illustrates the variations of q u of clean and silty sand with the initial stress anisotropy represented by η s .For each f c considered in this study, q u has an increasing and then decreasing trend as η s increases from − 0.4 to 0.6 and reaches its maximum value at η s = 0, indicating that the initial stress anisotropy has a detrimental effect on the strength of both clean and silty sand.This trend deviates somewhat from Kato et al. 62 , Georgiannou and Konstadinou 63 , and Pan et al. 54 , who reported that anisotropic consolidation with σ v > σ h at a lower η s level leads to an increase in the shear strength of sand.It is also shown that the addition of silty fines can either enhance or reduce the shear strength of sand, depending on f c .Specifically, the trend of q u against η s shifts upward as f c increases to 10% and then moves downward dramatically with further increases in f c .Thus, the curves of f c = 10% and 20% constitute the upper and lower bounds, respectively.This signifies that there exists a critical f c between 10 and 20%, at which although most of the fines are confined within voids, some of them separate the coarse particles from one another, increasing the fragility of the soil 58,64 .This is essentially different from f c,th determined above.The significant differences in the strength characteristics between clean and silty sand under different anisotropic consolidation, as shown in Fig. 12, are intimately related to the particle packing and arrangement of the mixtures.Through the discrete element method (DEM), numerous studies have been performed to investigate the influence of fines and stress anisotropy on the structure and mechanical properties of granular materials (see Minh et al. 65 ; Shire et al. 59 ; Zhou et al. 66 ).It is recognized that grains under gravity may cause their long axes to orient horizontally, making the sample become weaker in triaxial extension than in compression.This may explain the decrease in shear strength for samples with an extensional static deviatoric stress (i.e., a negative η s ).When the sample is anisotropically consolidated with a comparatively higher (compressional) static stress, more grains tend to lie on the horizontal direction 54 , especially for the sample with crushed fines that have more elongated particles.Consequently, the silty sand under anisotropic consolidation features a higher degree of anisotropy than the isotropically consolidated sample owing to the preferential orientations of particles, leading to the former being more contractive and susceptible to shear failure.Nevertheless, a micromechanical study through direct grain-scale observations should be further undertaken to better understand the underlying mechanisms.
Under undrained triaxial loading, the soil stiffness is usually quantified in terms of the undrained Young's modulus, E u , which is defined as the secant slope of the deviatoric stress-strain curve 67 .In Fig. 13, variations in E u with the associated axial strain are plotted on a logarithmic scale for isotropically consolidated samples with varying f c .Overall, a considerable degradation in E u is observed for the strain levels considered herein.As shown, the stiffness degradation curve of the clean sand is located on the lower part of the figure, whereas curves of f c = 5-20% are closely arranged on the upper portion.This indicates that the silty sand has significantly higher stiffness than that of clean sand under isotropic consolidation; however, this strengthening effect is insensitive to the amount of fines.It can also be observed in Fig. 13 that the difference in stiffness data between the clean and silty sand gradually narrows as strain increases, and the stiffness degradation curves tend to coincide when the axial strain surpasses 1%.
According to Clayton 67 and Pan et al. 54 , the secant stiffness at ε a = 0.1% is significant for analyzing soilstructure interaction since strain levels associated with geotechnical structures, including spread foundations, retaining walls, and tunnels, always fall within this range.Figure 14 shows a summary of the undrained Young's modulus E u at ε a = 0.1% (E u,0.1 ) for clean and silty sand under different consolidation conditions.Similar to Fig. 13, Fig. 14 reveals again that the addition of 5-20% silty fines to clean sand has a beneficial influence on the stiffness, irrespective of η s levels.However, the E u,0.1 values for silty sand samples under specific consolidation condition do not display a systematic shift as f c transitions from 5 to 20%, indicating that the effect of f c on stiffness is non-monotonic.It is also shown that both the clean and silty sand have an overall decreasing trend of E u,0.1 as η s increases from − 0.4 to 0.6 with two exceptions, which are samples with f c = 0 and 10% subjected to η s = 0.6.As a result, it is observed that anisotropic consolidation with σ v < σ h (η s = − 0.4) enhances the soil stiffness during triaxial compression in comparison to isotropic consolidation (η s = 0), whereas anisotropic consolidation with σ v > σ h (η s = 0.4 and 0.6) has an adverse effect.These findings are consistent with those reported by Yamashita et al. 68 and Pan et al. 54 , who discovered that static deviatoric stress on one side (compression or extension) might reduce the stiffness when the sample is subsequently sheared on the same side but strengthen it when the sample is sheared on the other side.

Conclusion
A series of undrained triaxial compression tests were conducted on clean and silty sand with the fines content f c ranging from 0 to 20% under either isotropic or anisotropic consolidation to an initial mean effective stress of p 0 ' = 100 kPa.The adequacy of state indices, such as the global void ratio e, relative density D r , and state parameter ψ in characterizing the behavior of isotropically consolidated samples was examined.The results of tests considering the effect of initial stress anisotropy under a constant ψ (= − 0.03) condition were presented and analyzed, with a focus on the strength and stiffness characteristics.The following main conclusions can be drawn from this study: (1) Comparing the undrained shear behavior of clean and silty sand using different state indices reveals varying perspectives on the effect of silty fines.While substantial differences in soil behavior under constant e or D r are evident, the similar overall behavior with a constant ψ implies that ψ is a comparatively superior predictor for synthesizing the responses of sand-fines mixtures, which also provides fairly good correlations to the instability stress ratio.(2) The state of axial strain developed to 25% is used to approximate the critical state of mixtures.A unique critical state line is obtained in the q-p' plane for samples mixed with 5-20% fines, yielding a critical state friction angle of 32.1°, which is slightly larger than that determined from the clean sand.This is mainly due to the presence of crushed fines that are more angular with irregular geometry than the host sand.Conversely, the critical state locus in the e-logp' plane is non-unique and can be described with a power function, exhibiting a declining tendency as f c increases up to 20%.(3) Under a constant ψ condition, the silty sand with 10% f c has the strongest dilation and highest shear resistance.A comparably higher static stress ratio η s typically leads to a deterioration in the monotonic shear behavior.Specifically, the undrained shear strength increases and then decreases as η s increases from − 0.4 to 0.6 and peaks at η s = 0, indicating that the initial stress anisotropy has a detrimental effect on the strength of both clean and silty sand.A grain-scale interpretation is made to provide a better understanding of these macro-observations; however, further laboratory experiments covering a wide range of f c and ψ and micromechanical studies through discrete element simulations are favorably undertaken to examine the underlying mechanisms.(4) The initial stress anisotropy may act positively or negatively on the undrained shear stiffness quantified by the secant Young's modulus, depending on the direction of η s .The stiffness of mixtures in triaxial compression may reduce if the sample undergoes a compressional η s but increase if it acts on the extension side.Moreover, the addition of 5-20% silty fines to clean sand has a favorable impact on the stiffness, regardless of the η s levels considered herein.

Figures 2
Figures 2 and 3 illustrate the effective stress path and stress-strain curve, respectively, of isotropically consolidated samples with f c = 0, 5%, 10%, and 20%.These data show the effect of the global void ratio e after consolidation on the undrained shear behavior.As shown in Fig.2a, the clean sand with e = 0.749 shows predominate

Figure 1 .
Figure 1.Particle size distribution of the test materials.

Figure 4 .
Figure 4. Comparisons of responses of isotropically consolidated samples with a constant relative density or void ratio: (a,c) effective stress paths; (b,d) stress strain curves.

Figure 5 .
Figure 5. Critical state lines in (a) q-p′ plane and (b) e-p′ plane.

Figure 6 .
Figure 6.Comparisons of responses of isotropically consolidated samples with a constant state parameter: (a) effective stress path; (b) stress strain curve.

Figure 12 .
Figure 12.Undrained shear strength of clean and silty sand under different consolidation conditions.

Table 1 .
Index properties of the test materials.